"Last night when our political team saw the CNN post saying that Dr. Carson was not carrying on to New Hampshire and South Carolina, our campaign updated grassroots leaders just as we would with any breaking news story," Cruz said in a statement first shared with CNN. "That's fair game. What the team then should have done was send around the follow-up statement from the Carson campaign clarifying that he was indeed staying in the race when that came out."

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Carson said Tuesday he accepted the apology, but questioned whether there was a deeper "cultural issue" with Cruz's campaign.

"As a Christian I will accept the apology but it doesn't correct the problem," Carson told CNN. "This is a cultural issue when people in your campaign feel that it's ok to distort the issues to their political advantage and to tell absolute lies. And the question really is will there be any consequences for that."

Republican National Committee Chairman Reince Priebus told Tapper on "The Lead" Tuesday that he would not get in the middle of the fight between Cruz and Carson.

"Look, for me to get in the middle of litigating these things that are going on behind the scenes, that we find out about the next day -- I think it was explained by one of the campaigns as to what happened, I think you take them at their word for it and we move onto the next state," Priebus said.

But the Cruz campaign reacted to the news by incorrectly telling precinct captains -- while the caucuses were still taking place -- that the move signaled Carson would be dropping out of the race.

Twenty minutes after the caucuses began in Iowa, Rep. Steve King, the Cruz campaign's national co-chair, retweeted CNN's Chris Moody about Carson's break from the campaign trail before New Hampshire's primary.

"Carson looks like he is out. Iowans need to know before they vote. Most will go to Cruz, I hope," King tweeted. In another tweet, the congressman said Carson's next steps were "equivalent of suspending."

Carson looks like he is out. Iowans need to know before they vote. Most will go to Cruz, I hope. https://t.co/lW5Js50EMA

Cruz Iowa staffer Spence Rogers also suggested in an email to precinct captains that Carson may be announcing the end of his campaign next week.

"Breaking News. The press is reporting that Dr. Ben Carson is taking time off from the campaign trail after Iowa and making a big announcement next week," the email read. "Please inform any Carson caucus goers of this news and urge them to caucus for Ted Cruz."

Carson said Tuesday that he merely wants to sleep in his own bed before heading back out on the trail.

"Anybody who's been on the road for almost three weeks recognizes exactly what I'm talking about -- you need some fresh clothes. It's not like you're going to be there forever," Carson told CNN. "It's very nice to sleep in your own bed every now and then. And if anybody can't understand that I feel sorry for them."